There are a few important precursors to this work that deserve mention. The first is Hrušák's proof [1]} that the principle \(\diamondsuit _\mathfrak {d}\)  implies that \(\mathfrak {a}\) , the minimum cardinality of an infinite mad family of subsets of \(\mathbb {N}\) , is \(\aleph _1\) , and later [2]} that \(\diamondsuit (\mathbb {N}^\mathbb {N},\lnot >^*)\) , also known as \(\diamondsuit (\mathfrak {b})\) , suffices. Since \(\mathfrak {b}\le \mathfrak {a}\) , this determines the value of \(\mathfrak {a}\)  in all of the aforementioned canonical models.
